Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

The Trefftz plane — where it appears

A plane drawn far downstream of a wing, across which the wake's vorticity is counted. The induced drag is an integral over that plane alone, which is why two wings with the same wake have the same drag whatever their shapes.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

What the ground actually gives a wing. Induced drag near the ground as a fraction of the same wing's in free air, against height in spans, at constant lift. The ground is a plane of symmetry, so the wake is joined by a mirrored wake of opposite circulation below it, and the upwash from that image is what takes the drag away. At a tenth of a span the wing keeps 0.516 of its induced drag — a saving of 48 per cent — and by a span and a half the effect is 1.3 per cent and going. The lift is held fixed all the way along this curve: what the ground gives here is not more lift, it is the same lift for less drag, and the two are different claims about a landing aeroplane.

The cushion that is not there

Ground effect is usually explained as air trapped under the wing and compressed into a cushion. At the heights an aeroplane actually flies at, the air under the wing is moving at four fifths of flight speed and most of what the ground gives is a drag saving the story never mentions.

misconceptions · Ground cushion Five quarters of the span, the same bending moment, 64/75 of the drag. The bell-loaded wing's induced drag and root bending moment against its span, both as fractions of an elliptic wing of unit span carrying the same lift. At equal span the bell is worse: its span efficiency is exactly three quarters. But its bending moment is exactly four fifths, and bending moment grows in proportion to span while drag falls as its square — so there is a span at which the bell has bought back the structure and is ahead on drag. It is at exactly five quarters, where the two curves are at 1 and 0.8533. Both numbers are rational and neither was put in by hand.

The loading nobody used

Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.

circulation · Bell shape The wake, and the velocity it gives itself. The cross-section of the wake far behind a wing with turned-up tips, with the velocity the wake induces on itself drawn as arrows normal to the trace. Induced drag is the integral of the circulation against that velocity and nothing else — this picture contains the entire quantity. It also contains no information whatever about where the surfaces were: two wings a chord apart and two wings ten chords apart produce the same picture and therefore the same drag, which is Munk's stagger theorem stated as a fact about what the arithmetic can see.

A wing that leaves the plane

A winglet is not a fence and it does not block anything escaping round the tip. The induced drag of any system of lifting surfaces depends on one cross-section of its wake and on nothing else whatever, and a wake that reaches upwards is cheaper for the same reason a wake that reaches sideways is.

circulation · Winglet Four aeroplanes, one number. Four biplanes with the same gap and the same loadings, staggered by nothing, by four tenths, by nine tenths and by one and six tenths of a chord. Their induced drags agree to the last bit of the arithmetic — the calculation cannot even express the stagger, because the Trefftz plane is a cross-section and everything drawn here projects onto the same one. That is Munk's stagger theorem, and stating it as the drag is unchanged understates it: there is no place in the computation where the stagger could be entered.

Two wings and it does not matter where

Move one wing of a biplane a chord forward and the induced drag does not change. Not approximately, not to a good approximation — the calculation that gives the induced drag has nowhere to put the stagger, because everything projects onto the same cross-section of the wake.

circulation · Biplane Tip to tip, two wings cost exactly half of what they cost apart. The induced-drag saving of a pair of wings against the gap between their tips, as a fraction of what the two pay flying alone. With the tips touching the pair is one wing of twice the span carrying twice the lift, and the arithmetic of that is exact: the drag halves, and the computation returns 0.499257 of the separate figure. Pull them apart and the saving falls away with the square of the distance. Birds fly in a V because the tips are the part worth overlapping, and the spacing that pays is a small fraction of a span rather than any distance a formation could hold by eye.

The lift beside a wing

Fly two aeroplanes with their wingtips touching and the pair costs exactly half what the two cost apart. Not approximately half — the arithmetic is a closed form, because two wings tip to tip are one wing of twice the span, and induced drag goes as the square of it.

circulation · Formation A closed wake, and the loading of least drag on it. The wake of a box wing in the plane that decides its drag, with the circulation of least induced drag drawn as a thickness along it. The horizontal members carry a loading close to elliptic and the vertical ones carry a share that lifts nothing — they contribute no lift, since lift is Γ dy and dy is zero on a vertical, and they change the drag by changing where the wake's vorticity is. At a gap of 20 per cent of span this system costs 67.1 per cent of what a single wing of the same span and lift would.

A wake that closes on itself

Prandtl's best wing system is a rectangle, not a wing. Solving for the loading on a closed wake turns up a circulation that costs nothing and does nothing — a gauge freedom in the middle of an optimisation — and a drag that keeps falling with no floor under it.

circulation · Box wing The wall is a boundary condition, solved for rather than reflected. A model's trailing vortices in a closed working section, with the sheet of sources that makes the wall a wall drawn as a displacement of the wall itself. The interference is an upwash — the model looks better than it is, and the correction factor comes to 0.1242 against the exact value 0.1250 that the image at the inverse point gives for a circle. The arrows are the interference velocity alone, with the model's own downwash removed, drawn to a common scale set by the longest of them.

The walls are in the answer

A wind tunnel measures a wing in a box the aeroplane will never fly in, and the box is worth a fifteenth of the induced drag. The correction is exactly an eighth for a closed circular section and exactly minus an eighth for an open jet, and the sign is the whole argument.

circulation · Tunnel interference How flat the optimum is. The same curve near its maximum, with the bands within a tenth and a half of one per cent of the best shaded. Every taper ratio from 0.31 to 0.42 is within a tenth of a per cent of the optimum, and the whole band from 0.25 to 0.51 is within half a per cent. The optimum is exact, and choosing it rather than its neighbour buys nothing a wing can measure.

The optimum that does not matter

Elliptic loading gives the least induced drag, exactly, and the proof is one line: the penalty is a sum of squares. Which is also why the optimum is flat enough that a quarter of the design space sits inside a tenth of a per cent of it.

circulation · Taper What every wing pays to bend its root less. Least induced drag against the root bending moment of the lift, both as fractions of the elliptic monoplane of the same span and lift, for the monoplane and for box wings with gaps of a tenth, a fifth and two-fifths of the span. Each curve is a parabola with its minimum at that wing's unconstrained optimum. The box wings' minima sit to the right of the monoplane's — they load their lift further out — and their parabolas are shallower: the gap-of-a-fifth box can bend its root 21 per cent less than the elliptic monoplane and still match its drag.

A lighter spar turns a box wing into a biplane

Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.

circulation · Box wing

Named alongside it

The objects these essays reach for when they reach for this one.

Induced dragCirculationSpan loadingDownwashOptimisationModel limitMunk's stagger theoremNon planar wakeBending momentBoundary conditionConstraintLift coefficient

All concepts